This report studies the application of spectral domain analysis to the problem of the high-frequency diffraction of electromagnetic waves. The basic formulation of the spectral approach is first summarized and then applied in a unified fashion to the important problems of the non-uniform illumination of a half-plane and staggered parallel planes. Special attempts are made to analytically determine the fields at the shadow boundaries and numerically evaluate them in the transition regions. Comparison is made with other high-frequency asymptotic techniques and some unique conclusions are obtained. Furthermore, it is shown that the important task of testing and improving high-frequency solutions can be accomplished by employing the Fourier transform of the integral equation for the surface current and using Galerkin's method in the spectral domain. Some examples of this approach are also included.